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# Contents

## Definition

Given a topological space $X$ equipped with a continuous function $f \,\colon\, X \longrightarrow \mathbb{R}^1$ to the Euclidean space of real numbers, it’s Reeb graph is the quotient topological space of $X$ by the equivalence relation which regards two points $x, y \,\in\, X$ as equivalent iff they are in the same connected component of the same level set. Under good conditions this is a CW complex, necessarily 1-dimensional, and as such an undirected graph.

## References

Original article:

• Georges Reeb, Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique, Comptes Rendus Acad. Sciences Paris 222 (1946) 847-849 $[$crid:1571417125676878592$]$

Further developments:

• Łukasz Patryk Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold, Topol. Methods Nonlinear Anal. 52 2 (2018) 749-762 $[$arXiv:1805.06727, doi:10.12775/TMNA.2018.029$]$